1. Introduction

Angularly-resolved light scattering (ALS) is an optical technique sensitive to scatterers’ size and refractive index (RI), making it a useful tool for evaluating cellular changes during various biological processes, including disease progression. ALS measures the intensity distribution $I(\theta )$ scattered from a sample under plane wave illumination, where $\theta$ is the polar angle of deflection. Modelling the object’s underlying structure under certain assumptions, this intensity distribution provides relevant information about the size distribution of the scatterers. ALS has exhibited sensitivity to mitochondrial connectivity and metabolic states [1–3], red blood cell size [4], and organelle population sizes within cell ensembles [5,6].

In ALS-based size analysis, cell scattering is most commonly modeled with Mie theory, using the approximation that all scatterers have the same refractive index, are all spheres, and are immersed in a homogeneous medium [5–7]. These simplified cell models remove sources of variability present in real cells, such as RI variations in the cytosol, scatterers with different ranges of RI, and non-spherical scatterers, allowing for scatterer size distributions to be extracted [5–7].

A potentially significant application for ALS in biology is the label-free evaluation of single cells. Single-cell analysis has widespread applications in studying immune response [8], tumor cell differentiation [9,10], and drug development [9]. For these applications, however, the Mie theory-based assumptions for ALS can have significant impact on the accuracy of the recovered cells’ properties. This is especially true in single cells where there are fewer total organelles, and there is less averaging over non-spherical organelle orientations.

Other groups have validated organelle size estimates at the ensemble level, e.g. by measuring isolated organelles at numbers corresponding to the contents of many cells [3]. However, information is lost when only ensemble measurements are made; such methods do not provide the desired information regarding the organelle size distribution of a single cell [10]. Other methods, such as scanning electron microscopy, provide high-resolution single-cell measurements enabling accurate organelle sizing. This approach, however, requires cell fixation and long acquisition times making this approach ill-suited for rapid, label-free sample analysis. In 2009, Kalashnikov et al. [11] used tomographic phase microscopy (TPM) to obtain 3D RI maps, and compared tomography-based size estimates of the major and minor axes of the nucleus in an HT-29 cell to angular scattering-based estimates. They applied this sizing to large organelles such as the nucleus and nucleolus, but not to smaller organelles.

In this work, we assess the accuracy of the simplified cell model of spheres in a homogeneous medium for predicting the angular scattering from real single cells. Three-dimensional refractive index maps of single cells are obtained via optical diffraction tomography (ODT), as will be described below. We use the 3D fast Fourier transform (FFT)-based “Wolf transform” [12] to compute the angular scattering from each tomogram. We then enforce a series of model assumptions (e.g. spherical scatterers, homogeneous cytosol) by digitally manipulating the original refractive index map and again computing the angular scattering. In this way, we quantify how each successive simplification increases the discrepancy with the original scattering.

The ODT approach also provides a direct calculation of angular scattering using only normal-incidence planar illumination. Unlike the Wolf transform, this direct approach does not require the Born approximation of small refractive index variations. The two angular-scattering approaches, direct and Wolf-transformed, are compared. Examining the effects of all these model simplifications is important for understanding the potential of assessing non-nuclear organelle size distributions in single cells.

2. Background

2.1 Tomographic phase microscopy and optical diffraction tomography

TPM is a subset of quantitative phase imaging (QPI), a set of imaging tools and techniques recovering the relative phase delay and light scattering induced by the sample compared to the surrounding medium. This phase delay couples both the sample’s RI and thickness information into an optical path length providing sensitivity to morphological features of a biological sample. Conventional 2D QPI systems recover this optical path length, while 3D QPI systems utilize angle-scanning illumination and other phase-encoding strategies to obtain quantified 3D RI distributions [13]. For angle-scanning methods, the unique elements of the system are typically composed of a controllable illumination source, such as galvanometer mirrors, LED arrays, or digital micromirror devices (DMDs) [14], and an inverse model relating the acquired interferograms or intensity images to the object’s 3D structure. Early TPM implementations relied on filtered back-projection models similar to Computed Tomography systems [15] that did not properly account for diffraction effects from the object. Later TPM systems [16] adopted diffraction-based tomography models based on the theoretical work of Wolf’s 1969 paper [12]. Recently, more advanced approaches have modeled broadband illumination [17,18] and multiple scattering using the Beam Propagation Method [17] for more accurate sample reconstructions.

The models of angular scattering in this work assume the monochromatic approximation, which is further discussed in Section 5.1. We briefly review the monochromatic diffraction-based model based on Wolf’s work below.

The 3D diffraction-based tomography model describes the object as a scattering potential $V(\mathbf {r})$ composed of the RI variation between the object $n(\mathbf {r})$ and the surrounding homogeneous medium ($n_m$), i.e.

The scattering from the object is inherently nonlinear due to multiple interactions between the field and object, but this term can be approximated as linear using the first Born approximation when the object is sufficiently weakly scattering [12]. Using this approximation with plane wave illumination, the 3D Fourier transform of the object’s scattering potential $\tilde {V}(K_x,K_y,K_z)$ can be evaluated in terms of angular scattering at a particular $z$-plane $\tilde {U}_s(k_x,k_y;z)$ [19]. For an incident plane wave with unity amplitude, the scattering potential can be extracted from the scattered field using

Combining the mapped Fourier space information from multiple measurements of $\tilde {U}_s(k_x,k_y;z)$ under different illuminations expands the recovered 3D scattering potential distribution for more accurate 3D sample reconstruction. Obtaining full 3D Fourier space coverage at the origin would require physically rotating the sample or the system. Since most ODT systems are confined to a limited angle scanning range, they cannot sample the object’s full spectrum and thus lose low axial spatial frequency information (small $K_z$ values). This loss is commonly referred to as the “missing cone problem," and results in axial elongation artifacts in the reconstruction [20]. As shown in recent work, the effects of these artifacts can be minimized using iterative reconstruction algorithms incorporating nonnegativity, total variation, and other prior assumptions on the underlying object [16,17,21]. The success of this 3D reconstruction modality has resulted in multiple commercialized ODT systems for high-resolution 3D cell tomography [e.g., Nanolive (Switzerland) and Tomocube (Korea)].

2.2 Obtaining angular light scattering from ODT

The angular scattering from a cell under planar illumination can be modeled using a 3D Fourier transform of the scattering potential, combined with a scaling factor. As invoked once already, we will refer to this Fourier relationship defined in Eq. (3) and derived by Wolf [12] as the Wolf transform. Starting from a cell’s refractive index map provided by ODT, the Wolf transform can be used to compute the angular scattering $I(\theta )$ from that cell volume. From Eq. (3), at the $z=0$ plane

The 2D scattering intensity (left hand side of the equation) is obtained from evaluating the scaled scattering potential in $K$-space (right hand side of the equation). This is then averaged over voxels of constant polar angle $\theta$. The distance from the origin in $K$-space, or $K=\sqrt {K_x^2+K_y^2+K_z^2}$, is related to the angle $\theta$ using [22]

Equations (5) and (6) can then be used to compute $I(\theta$) by averaging over constant $\theta$.

3. Methods

3.1 Tomocube commercial ODT system

Data was collected on a commercial ODT system (Tomocube HT-2) with $\lambda =532$ nm. This system uses a programmable digital micromirror device to sequentially illuminate the sample at 48 different azimuthal angles (with a polar angle of 45$^\circ$) and one illumination at normal incidence. A fiber beam splitter is used to create a reference wave that interferes with the sample wave to create an off-axis interferogram at the detector for each illumination angle. Using all of the interferograms, the Tomocube software reconstructs a $844\times 844\times 210$ 3D array of the sample’s refractive index using an iterative reconstruction algorithm with a nonnegativity constraint. The lateral and axial resolution of the reconstructions are 95 nm and 190 nm, respectively. The acquired volume rate for this system is 2.5 Hz. This speed was found empirically to be adequate for evaluating the adhered cells with minimal motion artifacts. The interferograms are measured using a 4.8 $\mu$m pixel pitch camera, with a system magnification of 58.3. While the reconstruction algorithm is specific to the Tomocube system, this holotomography algorithm is expected to use a modified reconstruction algorithm based on the Fourier diffraction theorem with additional prior assumptions such as nonnegativity. Tomocube provided us with a custom script to enable access to the raw interferograms.

This work compares angular scattering analysis from the 3D tomogram reconstructions as well as that computed from the normal incidence interferogram. The tomogram reconstructions are digitally manipulated and Wolf-transformed to obtain the angular scattering, as described in Section 3.4.

3.2 Fourier transform light scattering

Separately, the normal incidence interferogram from the Tomocube dataset was used to compute each cell’s complex field using spatial filtering to remove the twin image and DC term [23]. The resulting QPI image was digitally converted to the angular domain via a 2D FFT, following the Fourier transform light scattering (FTLS) approach of Ding et al. [22]. Exploiting the high circular symmetry of forward scattering, the angular scattering intensity pattern was then azimuthally averaged to yield a direct measurement of $I(\theta )$.

3.3 Sample preparation

Living human embryonic kidney (HEK) cells (High Throughput Sciences Core, Koch Institute at MIT) were plated at two densities of $1\times 10^5$ and $0.5\times 10^5$ cells in 1 mL onto a Tomodish (Tomocube Inc., Republic of Korea). The Tomodishes were coated with 0.1% Poly-L-Lysine to promote cell adherence by washing the dish with 100% ethanol followed by distilled water, letting the dish dry, and then coating the dish with Poly-L-Lysine for 5 minutes before washing three times with distilled water. The cells were immersed in DMEM cell media (10% Hyclone cosmic calf serum, 1% non-essential amino acids, and 1% pen/strep). The DMEM media was measured on a refractometer to have a refractive index of 1.337, matching water’s RI. The cells were placed in the Tomodish 24 hours before use in the ODT system. Nearly no cell death was observed prior to measurement acquisition.

3.4 Angular scattering intensity of ODT-based cell models

As previously described, using Mie theory to estimate $I(\theta )$ requires making simplifying assumptions to describe cellular scattering. Having access to the full 3D refractive index map of the sample creates the opportunity to explore the impact of each model assumption on the scattering data. Starting with the full tomogram supplied by Tomocube’s software, we simplified the model progressively by digitally manipulating the cell’s refractive index in a systematic manner. We then investigated the impact on the angular scattering using a Wolf transform of the modified refractive index (RI) map, as was described in Section 2.2.

The five cell models used to explore the effect of model assumptions are summarized in Table 1 and in Fig. 1. Starting from the original tomogram, the first modification sets the RI of all extracellular voxels to the average of the cytosol to go from model (i) to (ii). This step removes cell shape-induced scattering signal and is discussed in Section 3.5. The second simplification, from model (ii) to (iii), thresholds the cell to define organelle regions and sets remaining intracellular voxels to match the cytosol RI. This simplification evaluates the condition where the only scatterers in the object are organelles in a homogeneous cytosol. This model is described in Section 3.6. The step from model (iii) to (iv) replaces each organelle with its area-equivalent sphere, as described in Section 3.7. This step enforces the assumption that all the scatterers are spherical in shape. The RI of each sphere is set to the mean RI of each organelle. For model (v), the scatterers are all homogenized to have the same mean RI from all organelles in the cell.

 

Fig. 1. (a) Conceptual diagram of cell models described in Table 1 including (i) original tomogram, (ii) RI-matched, (iii) homogenized cytosol, (iv) spherical organelles, and (v) spheres of same RI. (b) Maximum intensity projection of RI tomogram, with inset shown in (c) for each model. (d) Corresponding normalized angular scattering computed for each model.

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Table 1. Description of cell models.

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3.5 Whole cell scattering removal

The refractive index mismatch between a cell (cytosol $n\approx 1.37$) and its immersion medium ($n_m\approx 1.337$) has been experimentally demonstrated as a significant source of light scattering [24]. To investigate the effects of organelle scattering in isolation, it is desirable to eliminate this RI mismatch. One approach achieves experimental RI matching using iodixanol to raise the immersion fluid’s refractive index [24,25]. Alternatively, the 3D RI profile obtained via tomography could be used to digitally perform the matching process without modifying the cell’s immersion medium.

Experimental refractive index matching has been shown to reduce the contributions of whole-cell scattering [24]. However, the osmolality mismatch between the cell and the iodixanol solution could cause changes in the cell’s internal RI distribution. This can be ameliorated by changing the solution’s salt content but would require further experimentation to demonstrate that the new solution does not adversely affect the cell.

Digital refractive index matching is instead used in this work to computationally remove the effect of whole-cell scattering without potentially modifying the cell. Digitally index matching the media to the cell is enabled by access to the 3D RI map. In 2D complex field images used for FTLS, diffraction effects at the cell boundaries due to the cell’s 3D nature make it less straightforward to classify pixels as belonging to either the cell or media. Digital RI matching can be achieved in the tomogram by identifying all RI values less than a threshold value, and setting the index of those voxels equal to that of the cytosol (as determined below).

To find an appropriate value between the cytosol and cell media index, the Otsu thresholding method was implemented [26]. Figure 2(a) shows an axial slice of an HEK cell in DMEM media, while (b) shows the histogram of all voxels in the tomogram. Note that the threshold value identified with Otsu’s method separates the two peaks in the histogram; the lower index peak corresponding to the media, and the higher index peak corresponding to the cytoplasm. Otsu’s method was applied to each tomogram from 28 cells, and the mean and standard deviation of the 28 threshold values were found to be 1.350 and 0.002, respectively.

 

Fig. 2. (a) Tomogram’s axial slice of an HEK cell in DMEM media. (b) Histogram of tomogram’s voxel values, indicating peaks corresponding to media and cytoplasm. Dashed line indicates threshold used to segment cell. (c) Same axial slice after digital index matching, showing less contrast at cell boundaries. (d) Lateral slice after digital index matching, with axial blurring causing imperfect index matching.

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To find the average cytosol value, regions of an axial slice in the middle of a cell were manually selected that were within the cell’s boundaries and did not contain high-index scatterers. The mean cytosol value over 28 cells was found to be 1.364 with a standard deviation of 0.004.

The algorithm for digital index matching is shown in Table 2. The RI map is thresholded and the largest connected object is used to identify the voxels belonging to the cell media. Identifying the largest connected object and then inverting this to obtain the cell mask is important for leaving any vesicles inside the cell with a low RI (e.g. water-containing vesicles) unmodified. Identifying connected spaces below the threshold identifies only the extracellular media, while leaving the cell’s interior untouched. Voxels belonging to the cell are set to the original RI value, and voxels belonging to the media are set to 1.364.

Table 2. Digital refractive index matching algorithm

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One challenge with digital RI-matching is that the missing cone problem causes axial blurring, artificially lowering the RI in some regions of the cell reconstruction [17]. This means that a global RI match will not perfectly match the boundaries of the cell at all axial depths, resulting in residual cell-boundary scattering when conducting the Wolf transform of the modified RI map. This imperfect matching at all axial depths is visible in the lateral slice shown in Fig. 2(d), where the extreme axial regions of the cell have a visibly reduced RI compared to the media.

3.6 Cytosol homogenization isolating organelle scattering

Another assumption from prior models used to fit organelle scattering is that the scatterers are immersed in a homogeneous medium with a constant RI [5]. This implies that only the “high-refractive index” (defined by some threshold) organelles contribute scattering, and that the cytosol is homogeneous. Similar to the digital refractive index-matching of the cell, this cytosol homogenization can be digitally modeled by applying a threshold to isolate the organelles, and setting all voxels less than the threshold equal to the cytosol index that we previously defined ($n_m=1.364$).

An important issue to consider regarding this step is that the scattering will be sensitive to the choice of the threshold value chosen to identify the organelles. Otsu’s method is not as easily applicable because the high-index organelles are represented by too few voxels compared to the rest of the cytoplasm. Furthermore, the threshold chosen will affect the starting point for the algorithm described in the next section for segmenting organelles and replacing them with their area-equivalent spheres.

Existing angular scattering literature provides some context for selecting thresholds in order to segment organelles. The refractive index of scatterers has been modeled as 1.40 [5,6] or 1.39 [7] in a medium of 1.38 [5,6] when modeling scattering from intact cells. Others have cited that organelles’ refractive indices vary between 1.38 and 1.41, and that cytosol values typically fall between 1.36 and 1.375 [27]. Since this work aims to explore the validity of assuming that scattering in cells is from organelles with RIs between 1.38 and 1.4, we chose a threshold of 1.385 for organelle segmentation (described in more detail in the next section). Although there could be non-homogeneous regions within the cell that have lower refractive indices than this threshold, we are investigating the effects of assuming that these lower RI features do not strongly contribute to the scattering.

Since this homogenization model is based on a simple thresholding step, there is a continuous, threshold value-dependent variation in the resulting scattering between the RI-matched cell model and the homogenized cytosol. Since any conclusions drawn about the validity of assuming a homogenized cytosol are sensitive to this choice of refractive index, we also show scattering plots in Fig. 7 for a range of threshold values from 1.37 to 1.385 on an example cell.

3.7 Organelle replacement with area-equivalent spheres

Spheroidal scatterers are known to scatter like their area-equivalent spheres in the angular ranges considered in this work [28]. Since we are comparing to a Mie theory-based size estimate that assumes spherical scatterers, it is necessary to approximate organelle size with a characteristic diameter of the area-equivalent sphere. The segmentation process for assigning area-equivalent sizes to each organelle is shown in Fig. 3 and is described in the following steps:

 

Fig. 3. Algorithm for assigning sizes to each organelle, as described in Table 3. 1) Threshold to identify high RI regions. 2) Use watershed transform to separate touching organelles. 3) Identify axial slice corresponding to weighted centroid of each organelle. 4-6) Threshold at the refractive index contour line with the largest 2D gradient. 7) Assign size corresponding to area-equivalent sphere.

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Table 3. Algorithm for identifying area-equivalent spheres, with flow diagram depicted in Fig. 3.

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Apply a threshold to identify organelles, setting any voxels less than the threshold value of 1.385 equal to zero.

Apply the watershed transform. While thresholding the refractive index is useful for identifying regions where organelles are present, there are cases where two organelles may be connected in the 26-connectivity definition. All voxels belonging to either organelle would then be identified as a single connected object. Algorithms for separating touching objects often use a watershed transform, which operates by finding the boundaries between local minima, known as “catchment basins.” An important step to using the watershed transform is to transform the image into something with catchment basins (i.e., local minima) at the center of each object [29]. The distance transform is often used to obtain local minima at the center of objects; for our case, the inverted refractive index ($1-n$) creates catchment basins, because the small, non-nuclear organelles under study tend to have local maxima in refractive index at their centers.

Identify the axial slice for each organelle’s weighted centroid. In ODT reconstructions, the axial resolution is typically lower than lateral resolution due to the low axial spatial frequency coverage. This means that structures can appear artificially elongated in the z dimension and even have underestimated RI due to the blurring [17]. Since these axial spatial frequencies are fundamentally missing and we are already assuming spherical scatterers, a straightforward solution is to use an axial slice to estimate cross sectional area of the scatterer [30]. This avoids the problem of overestimating the axial dimension of the scatterer due to the blur. It also speeds up the size estimation, since the segmentation can now be done in 2D instead of 3D. While it is possible that some organelles might be longer in one dimension, it should on average be a good approximation to look at an axial slice if the organelles are randomly oriented. The 2D slice was upsampled by a factor of four to reduce pixellation effects in size estimation of small organelles. An example of an axial slice is shown in step 3 of Fig. 3.

Identify the iso-refractive index contours. To avoid making organelle size estimates highly sensitive to the original choice of refractive index threshold, and since different organelles have slightly different average refractive indices, each organelle’s axial slice was individually segmented to estimate cross-sectional area. Lines of constant refractive index, or iso-RI contours, are used to find the border of the organelle.

Compute the 2D gradient and threshold. The edge of each organelle is determined by identifying the iso-RI contour with the largest 2D gradient magnitude (i.e., where the refractive index is changing the most sharply) using Matlab’s “imgradient” function and the Sobel gradient operator [31]. The value of this iso-RI contour is used as a threshold to identify that organelle’s 2D extent.

Compute the area of the area-equivalent sphere by counting the number of pixels in the segmented organelle slice. This area is used to compute the diameter $D$ of the organelle’s area-equivalent sphere, using

Create the area-equivalent sphere with a RI set to the mean of the organelle’s voxels, and center the sphere at the location of the organelle’s weighted centroid (weighted by the RI value). The same matrix dimensions as in the tomogram reconstructions are used to represent a discretized set of spheres at different locations. For model (iv) in Table 1 and Fig. 1, the RI of the sphere is set to the mean RI of that organelle. For model (v), the RI of all the spheres is set to the mean RI of all the organelles in that cell.

3.8 Angular scattering error analysis

To quantify the difference in angular scattering, we define a percent difference metric $d$ between two angular scattering curves $I_1(\theta )$ and $I_2(\theta )$ averaged over all angles as

4. Results

4.1 Mie scattering and Wolf transform scattering comparison

Comparisons between the Wolf transform-computed scattering from the experimental RI map and the angular scattering computed via FTLS are provided for an HEK cell without digital RI-matching in Fig. 4. The Wolf transform plot was computed from the 3D FFT of the experimentally measured scattering data, as described in Section 2.2. The FTLS plots come from normal-incidence angular interferograms, as described in Section 3.1 and Section 3.2. The plots have close agreement in shape, although there are some discrepancies, most notably that the Wolf scattering is higher at the lowest scattering angles (around 1 degree). We attribute this discrepancy to two factors: 1) the lower object resolution in the single-illumination FTLS transform when compared to the multi-angle illumination Wolf transform and 2) the susceptibility of the Wolf transform object to reconstruction errors or noise, for example the missing cone problem as described in Section 2.1.4.

 

Fig. 4. Comparison of experimentally measured Wolf scattering (thick line) and FTLS scattering (thin line) for an HEK cell in DMEM media without digital RI-matching.

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The overall agreement between the two scattering plots suggests that the Wolf transform’s assumption of the first Born approximation is acceptable for these single cells. Furthermore, the supplemental material compares the Wolf transform of organelle-like spheres to Mie theory, to confirm the close agreement between the two theories.

4.2 Model assumption effects on single cell scattering

Five different cell models have been described, namely, the original tomogram as measured by the Tomocube instrument, a digitally refractive-index-matched cell, a cell with homogeneous cytosol and high index scatterers, a cell with spherical scatterers, and one with spheres of identical RI. Figure 5 shows maximum intensity projections of 3D data from the same HEK cell tomogram for each of the five cell models discussed in Section 3.4. Figures 5(a)–5(c) are maximum intensity projections along the z axis with the same colorscale, allowing visualization of the high-index organelles. Figures 5(d) and 5(e) show the area-equivalent spherical scatterers placed at the corresponding spatial locations. In Fig. 5(d), each sphere was assigned an RI matching the average of all voxels identified to that scatterer. In Fig. 5(e), all spheres were assigned the same RI, which is the mean index of all the scatterers in Fig. 5(d).

 

Fig. 5. Maximum intensity projection of (a) original tomogram, (b) refractive index-matched tomogram and (c) tomogram with homogenized cytosol. (d) Corresponding depiction of area-equivalent spheres with individually computed refractive index, and (e) the same mean refractive index. White arrows in (b) indicate some contiguous regions with RI larger than their immediate surroundings but lower than the threshold chosen for organelle classification.

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The corresponding angular scattering intensity for each of the models in Fig. 5 is shown in Fig. 6. In Fig. 6(a), the plots are normalized following Eq. (9) to have the same total scattering between 25 and 40 degrees. With this normalization, the five angular scattering profiles overlap closely at angles above 20 degrees but diverge at lower angles. Table 4 shows the average percent difference $d$ as defined in Eq. (8) between the normalized scattering from consecutive models. The largest difference ($d=135{\%}$) is from RI-matching. In Fig. 6(b), the same plots are left unnormalized to enable comparison of the relative total amount of scattering from each model. Plots 6(c) and 6(d) show histograms of the RI and area-equivalent diameter for the 138 scatterers identified in the cell. Note that the organelle diameters are mostly below one micron, and the refractive index peaks below 1.39.

 

Fig. 6. Scattering for the same HEK cell as the previous figure for all five versions (original, RI-matched, homogenized cytosol, spheres, and spheres of the same RI.) Normalized to have the same amount of scattering between 25 and 40 degrees in (a), and unnormalized in (b), both with semilog axes. Histograms of (c) refractive index and (d) diameter for the 138 area-equivalent spheres used to model the same cell.

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Table 4. Scattering percent differences ($d$) for example cell with different cell models.

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Figure 7 and the results in Table 5 demonstrate the effect of varying the chosen RI threshold value for defining organelles, which was held to be 1.385 for Fig. 5(c). Figures 7(a)-(c) show the homogenized cytosol maximum intensity projections for lower thresholds of 1.38, 1.375, and 1.37. The corresponding scattering is shown in (d), normalized to the total scattering between 25 and 40 degrees. The RI-matched case of 1.364 (c.f. Section 3.5) is also shown for comparison. Note that the number of larger scatterers increases as the RI threshold is lowered, and that this creates an increase in scattering at angles less than 15 degrees. This is consistent with the fact that large scatterers have more forward-peaked scattering (i.e., more scattering at small angles) compared to smaller scatterers.

 

Fig. 7. Maximum intensity projection images of an HEK cell’s tomogram, with voxels set to 1.364 for value below a threshold of (a)1.38, (b), 1.375, and (c) 1.37. White arrows in (c) indicate low-RI organelles. (d) Scattering from the corresponding tomograms, as well as with the threshold at 1.385 (Fig. 5(c)) and for the RI-matched case (Fig. 5(b)).

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Table 5. Scattering percent difference with different organelle thresholds.

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Out of the 28 cells that were measured, the percent differences for 17 of them are shown in Fig. 8 and in Table 6. The first scatterplot corresponds to the percent difference between the original and RI-matched tomogram; the second to the difference between the RI-matched and the homogeneous cytosol; the third between the homogenized cytosol and the spheres of different RI, and the fourth between the spheres of individual RI and the same RI. Note that the index-matching step causes the greatest source of difference in scattering ($d=150{\%}$), and the change from different index spheres to spheres of the same index caused the least ($d=3.8{\%}$).

 

Fig. 8. For all seventeen cells, percent difference in scattering (averaged over angles) between: (i) original tomogram and (ii) digitally RI-matched tomogram, (ii) RI matched and (iii) homogenized cytosol, (iii) homogenized cytosol and (iv) spheres of homogeneous RI, and (iv) spheres of homogeneous RI and (v) spheres of the same RI.

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Table 6. Mean scattering percent differences over 17 cells for different cell models.

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The cells that were not used were excluded for one of two reasons:

As described in Section 3.6, we chose a threshold value of 1.385 to isolate organelles. This was motivated by the desire to test assumptions in the literature that scattering from intact cells can be modeled as scattering from spherical organelles with a constant RI exceeding 1.38. Occasionally, the segmentation algorithm used to identify area-equivalent spheres would output a large ($>1\mu$m) scatterer that did not visually correspond to a scatterer of that size at the same location in the original tomogram. Four of the 17 cells required manual removal of between two and five organelles exhibiting this behavior. This was done to prevent the large scatterers, which were perhaps artifacts of an imperfect segmentation process, from dominating the computed scattering. Scatterers greater than 1 $\mu$m in diameter that did visually correspond to a scatterer in the original tomogram were left unmodified.

5. Discussion

5.1 FTLS and Wolf scattering comparison

The underpinnings of Mie theory, FTLS, and the tomogram reconstruction and analysis require various levels of assumptions about the sample and illumination. Mie theory, FTLS, and the Wolf transform all assume a monochromatic source, whereas the LED in the Tomocube instrument has a finite bandwidth. It is often sufficient to model LED illumination scattered by a sample with a monochromatic approximation at the LED’s center wavelength. This has for example been validated by comparing measured scattering from spherical beads to monochromatic Mie theory [32].

Mie theory is the complete vector solution to a sphere’s scattering, whereas FTLS and ODT reconstruction use a scalar approximation. Both the tomogram reconstruction algorithm and the Wolf transform used in scattering analysis additionally utilize the first Born approximation, which requires weak sample RI contrast. FTLS, which involves Fourier transforming the sample’s complex field, does not require this assumption.

To justify the use of the Wolf-transform to approximate scattering from cells, Fig. 4 compares the experimentally measured Wolf and FTLS-computed scattering from an HEK cell. The close agreement between the two curves confirms that it is reasonable to use FFT-based analysis of tomogram data, as was done in this work, to draw conclusions about the scattering from different cell models. To confirm that Mie theory also agrees closely with the Wolf transform, and to investigate the potential sampling artifacts, the supplemental material compares simulations of scattering from organelle-like spherical scatterers using Mie theory, a digitized Wolf transform of a sphere, and an analytical Wolf transform.

5.2 Comparison of cell models’ angular scattering

One goal of this work is to understand the limitations to how well the idealized cell model, which assumes spheres in a homogeneous cytosol, is representative of a real cell’s scattering. By breaking down the idealized cell model into a series of assumptions that can be enforced in a cell’s tomogram, we can identify to what extent each assumption changes the shape of the cell’s angular scattering pattern. The results shown in Figs. 6 and 8 lead to several takeaways. First, in almost all cells, the refractive index-matching step accounted for the largest percent difference. This is seen by RI matching causing the largest percent difference of 150% in Table 6. Similar to the findings in Dunn et al. [24], Fig. 6(a) shows the discrepancy in scattering from RI matching persists up to approximately 10 degrees. Since RI-matching can be achieved experimentally, this discrepancy in scattering does not by itself mean that the ideal cell model is unusable.

Cytosol RI heterogeneity is responsible for the next largest source of discrepancy between the ideal cell model and real cell scattering, $d=91.1$% in Table 6 and the second column of datapoints in Fig. 8. We note, however, that this result is influenced by the threshold used to distinguish organelles from the cytoplasm, as was demonstrated by Fig. 7. If a lower threshold value is chosen (e.g. 1.37), more of the cell content is classified as organelles, so homogenizing the cytosol causes less of a change in the scattering profile, compared to the threshold of 1.385 that was used in most of the calculations.

This is visible in Figs. 7(a)-(c), where there are larger RI regions (indicated with white arrows in b) that are visibly higher than their local environment. These regions are not segmented as organelles when the RI threshold is 1.385. These regions fall into a grey area where they could either be classified as heterogeneity in the cytoplasm, or as low-RI organelles deserving of treatment as area-equivalent spheres in the remaining analysis. Since we are comparing these results to the models previously reported in the literature, we chose to classify these regions as part of the cytoplasm since they are below the refractive indices (1.38-1.41) typically assumed to represent organelles [5,7,27]. These regions are also significantly more challenging to successfully segment due to low constrast, so simply lowering the RI threshold and applying the same algorithm does not consistently identify these scatterers. Note that in Fig. 7, these structures are only visible in (c) when the RI threshold is lowered to 1.37, which is only slightly higher than the cytosol index of 1.364. Improving the segmentation algorithm to add the capability of identifying these larger, low-index scatterers would provide a more accurate assessment of the assumptions. Alternatively, using a labelling method such as fluorescence imaging in parallel to the tomogram imaging could help to identify certain organelles in the tomogram, so that organelle identification would not depend solely upon RI.

The choice of refractive index threshold for organelle identification also affects the next model simplification of creating area-equivalent spheres. As was previously mentioned, there were occasionally large cell regions identified by the segmentation algorithm (between 1.5 and 3 $\mu$m area-equivalent diameter) that were manually removed because they either did not visually correspond to organelles in the tomogram, or because they corresponded to the low-index scatterers previously discussed with RIs between 1.364 and 1.385. Since these scatterers were inconsistently identified, we instead classified these features as heterogeneous cytosol regions rather than as organelles for the purposes of this study.

Enforcing the assumption that scatterers are spherical contributed significantly less to the overall percent difference. This is shown in Fig. 6(a), where the homogeneous cytosol and the spheres scattering curves overlap significantly; in the smaller cluster of data in the third column of Fig. 8, and in the smaller value of $d=21.0$% in Table 6. However, this result could partially be due to the segmented organelles already having high sphericity. In the cell shown in Fig. 5 for example, the cross-sections of organelles used to determine area-equivalent spheres had a median minor to major axis ratio of 0.85 (where a ratio of one would denote a circular cross-section). It is therefore possible that less-spherical organelles in other cell types might cause a larger difference in the scattering, when compared to the ideal model of spherical organelles.

The step of making all spheres have the same refractive index caused the smallest difference in the scattering, evident by the smallest mean $d=3.8$% in Table 6 and in the fourth column of Fig. 8. This implies that for the cells studied here, it is a good approximation to assume a single refractive index when fitting angular scattering from organelles to Mie theory, as is asserted by Asano and Sato [28]. However, the conclusions drawn regarding area-equivalent spheres and spheres with constant RI might be different if the larger, lower refractive index scatterers (white arrows in Fig. 7(b)) were also segmented and treated as organelles; since these have a much different refractive index and size from the small, high-RI scatterers, treating organelles as all having one RI would likely have a large impact on the scattering.

The ideal cell model that others have used to fit the scattering from intact cells assumes that all scattering can be well described by scattering from a collection of spheres with the same refractive index in a homogeneous medium [5–7]. It is clear that there are limitations to this model, due to the changes in the scattering from imposing these constraints on the tomograms of single cells. However, it is possible that the ideal cell model, despite its limitations, can still be used to extract stable and relevant metrics about changes or differences at the single-cell level. An important next step in analysis of the cell models will be to include a comparison of a scattering-based size distribution estimate to the segmented organelle sizes. This has not been conducted yet since the comparison should be performed using index-matched data to remove the whole-cell scattering, and because of the issues with reconstructing the experimentally index-matched tomograms (see Section 3.5).

Furthermore, we note the model simplifications on scattering presented here are specific to the evaluated HEK cell line and may not translate to all cell types. While the organelle scatterers in these HEK cells are not fully characterized, they primarily exhibit sub-micron diameters, differing from the 1-4 $\mu$m scatterers that Wilson et al. report to be responsible for 85% of the light scattering in their work [5]. Mourant et al.’s analysis of AT3.1 cells attributed $\sim$95% of the light scattering to mitochondrion-sized scatterers [5,7]. The low-RI, non-organelle regions in this work could explain the discrepancy between the 1-4 $\mu$m scatterers from Wilson and Mourant and the sub-micron scatterers identified here. We also note that the 85% and 95% values represent the amount of scattered light, which is related to but not equivalent to the number of scatterers (since larger scatterers scatter more light). Both Wilson and Mourant reported many more sub-micron scatterers than mitochondrion-sized scatterers; the larger scatterers dominate the scattering because of their larger cross-sectional area. Nevertheless, caution should be used in generalizing conclusions that were made based on experiments from a single cell type; they may not apply to other cell types where other types of scatterers dominate. It is also of note that this cell type did not have an obvious nucleus in the RI tomograms, making it difficult to assess the nuclear contribution to the scattering.

6. Conclusions

We have demonstrated a 3D FFT-based approach for assessing how accurately a simplified cell model describes a real cell’s angular scattering. Such model simplifications have been used by others for inversion of elastic light scattering from single cells to extract organelle size information. We found that the whole-cell scattering has a significant impact ($d=150.0$%), in agreement with recent experimental measurements [24], and should therefore not be ignored when modelling cellular scattering. Heterogeneity in the cytosol presents the next most significant mismatch between the idealized model and real cells ($d=91.1$%).

These results are overall promising for the use of angular scattering to estimate scatterer size distributions. The difference in scattering when digital refractive index matching is applied underscores the value of refractive index matching in general if the goal is to determine organelle sizes from angular scattering. Furthermore, the assumption that organelles scatter like their area-equivalent spheres causes a relatively small amount of changes in the scattering. The most problematic assumption in the cell model is that the cytosol is homogeneous; however, this could be reduced in effect if a broader range of organelle types are successfully segmented, rather than just the small, high index scatterers.

This work sets the stage for a significant amount of further studies. First of all, improving the segmentation algorithm to enable identification of “low RI” scatterers (i.e., those indicated by the white arrows in Fig. 7(b)) would allow for a more accurate comparison of cell models, since organelles of all sizes and RI values could be converted to their area-equivalent spheres.

The inability to directly compare an experimentally index-matched cell tomogram’s scattering to the FTLS measurement eliminated the possibility of comparing angular scattering-based organelle size estimates data to the sizes estimated by tomgram segmentation. Since the current system could not reconstruct accurate tomograms under index-matching conditions due to the nonnegativity constraint, there was no tomogram segmentation from an index-matched cell. Using an alternative reconstruction technique (i.e. without the nonnegativity constraint) could allow for accurate tomogram reconstructions with index-matched cell media. Comparing scattering-based organelle size estimates to tomogram-based analysis could lend confidence to the inversion algorithm used in the scattering approach.

Finally, repeating the digital FFT-based tomogram analysis demonstrated here for index-matched data from many cell types would provide useful insight regarding the cell model assumptions. Heterogeneity in the frequency of various organelles between cell lines could make certain assumptions more valid for some cell types than for others. Investigating the accuracy of model assumptions across a variety of cell types is important for verifying the efficacy and stability of estimating organelle size distributions.